DYNAMICS AND KINEMATICS OF SUBMARINE CABLE 



1167 



where Sy and S2 are lengths along the cable from the touchdown point 

 to the ship. Also, from Fig. 24 



Zi + Vt = (/i2 - /ii)/tan 13 + X. . (41) 



^m^>^^<^^#s^^^^<<^^^^^^'^ 



Fig. 24 — Piano wire configurations at the beginning and end of a descent lay. 



Equations (39), (40), and (41), together with the general equations 

 (Section 3.6) 



(24) 



allow one readily to solve for the average error e in the piano wire indi- 

 cation of ship ground speed. The result is 



sin a + sin ^ — k sin a sin /3 



e = — 1 '. : 7T : ; ; — — i- 



(42) 

 sin {a + ^) — \ sm a sm ^ 



For the small values of a and ^ which normally occur during the lajdng 

 of the piano wire, the terms X sin a sin ,8 and k sin a sin /3 are negligible. 

 Hence, the average error e is thus very nearly 



sin a + sin ^ . /,q\ 



sin (a + p) 



which, as (30) indicates, coincides with the amount of fill which would 

 be required to lay downhill with the straight-line or zero touchdown 

 tension configuration. Equation (43) is in turn closely approximated by 

 (Section 5.1) 



a/3 



27' 



(■14) 



